On Variants of CM-triviality

TL;DR

This paper generalizes the concept of CM-triviality relative to a fixed collection of partial types, demonstrating implications for the internality of stable fields and the structure of finite Lascar rank groups.

Contribution

It introduces a new generalized notion of CM-triviality relative to invariant partial types and explores its consequences in stability theory.

Findings

Stable fields are internal to the family under the new condition.

Finite Lascar rank groups have a normal nilpotent subgroup with an almost internal quotient.

The generalization extends the understanding of canonical base properties.

Abstract

We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalizes one-basedness. We show that, under this condition, a stable field is internal to the family, and a group of finite Lascar rank has a normal nilpotent subgroup such that the quotient is almost internal to the family.